CHAPTER SIX - User Web Pagesusers.monash.edu.au/~jpwalker/thesis_pdf/chapter6.pdf · 2000-08-25 · 6.3 INITIALISATION PHASE The one-dimensional soil moisture and heat transfer model

Chapter 6 − Synthetic Study: 1D Soil Moisture Profile Estimation Page 6-1

CCHHAAPPTTEERR SSIIXX

6. SYNTHETIC STUDY: 1D SOIL MOISTUREPROFILE ESTIMATION

The first step towards estimating the spatial distribution and temporal

variation of soil moisture profiles, was the establishment of the profile estimation

algorithm (section 3.5) for a one-dimensional soil column. Previous approaches

for assimilating observations of near-surface soil moisture content into a

hydrologic model (Chapter 3) include: (i) the continuous Dirichlet boundary

condition, (ii ) hard-updating and (iii ) Kalman-filtering. However, there is no

consensus as to which approach is the most eff icient for soil moisture profile

estimation, or on the effect of observation depth and update interval. While

application of these assimilation schemes is not new, this chapter identifies: (i)

which of the above assimilation schemes is most eff icient for soil moisture (and

temperature) profile estimation; (ii ) the near-surface soil moisture observation

depth required for soil moisture profile estimation; (iii ) how frequently near-

surface soil moisture observations must be made; and (iv) the most important

aspects and appropriate form for the hydrologic model to be used in the spatial

soil moisture profile estimation problem. The continuous Dirichlet boundary

condition is used to ill ustrate the effects of updating with continuous near-surface

soil moisture measurements, as may be available from a single near-surface soil

moisture sensor at a weather station.

The governing equations for flow of heat and moisture through

unsaturated soil , and the equations relating microwave observations to soil

moisture content, are highly non-linear. Therefore the continuous Dirichlet

boundary condition and hard-updating assimilation schemes are simpler then the

Kalman-filter assimilation scheme, as they allow the non-linear problem to be

solved directly. However, with hard-updating, the only way in which near-surface

observations can be transferred to deeper layers is through propagation of the

near-surface states down the soil profile by the internal physics of the model.


6.1 SYNTHETIC DATA

This chapter uses synthetic soil moisture and temperature data generated

from the one-dimensional soil moisture and heat transfer model developed in

Chapter 5, to test the soil moisture and temperature profile estimation algorithm.

The same soil moisture and heat transfer model was used to both (i) estimate the

soil moisture and temperature profile from assimilation of near-surface

observations, and (ii ) generate the synthetic data sets used for evaluation of the

soil moisture and temperature profile estimation algorithm. This allowed testing

of the soil moisture and temperature profile estimation algorithm independent of

errors in measurement of the soil moisture and temperature profiles, soil

properties, and surface fluxes. Furthermore, using the same model to generate the

synthetic data as was used in the soil moisture and temperature profile estimation

algorithm, eliminated the effects from model errors deriving from neglect of the

effects of hysteresis, thermal gradients, heat of wetting and vapour components of

the soil heat and moisture balance. The use of synthetic data also allows a broader

range of soil and climatic conditions to be investigated than for field data, and

allows for greater generalisation in the conclusions.

The soil moisture and heat transfer model developed in Chapter 5 was

used to generate 40 days of “ true” soil moisture and temperature profiles using the

van Genuchten (1980) moisture retention and hydraulic conductivity

Table 6.1: Soil parameters used in evaluation of the soil moisture and temperature profileestimation algorithm.

Total Soil Depth 100 cm

Number of Nodes 31

Soil Thermal and Hydraulic Parameters Clay Loam

Ks = 25 cm day-1

φ = 0.54

θr = 0.2

η = 0.008

n = 1.8

Proportion of Quartz = 0.03

Proportion of Other Minerals = 0.41

Proportion of Organic Matter = 0.02

Initial Conditions 20°C, −50 cm Matric Head


relationships. The soil parameters used by the soil moisture and heat transfer

model were those used by Entekhabi et al. (1994), Table 6.1.

The “true” soil moisture and temperature profiles were generated by

initialising the soil moisture and heat transfer model with the initial conditions of

Entekhabi et al. (1994), −50 cm matric head and 20°C uniform throughout the 1 m

deep soil profile, and subjecting the soil profile to the boundary conditions of

Entekhabi et al. (1994), 0.5 cm day-1 evaporation and a diurnal soil heat flux of

400 langley day-1 amplitude at the soil surface (Figure 6.1). The boundary

condition at the base of the soil column was zero soil moisture and heat flux.

6.2 KALMAN-FILTER OBSERVATION EQUATION

As thermal infra-red remote sensing observations only provide information

about the soil skin temperature, only the soil temperature of the surface node in

the soil profile discretisation can be updated. In contrast, remote sensing

observations of soil moisture are related to the soil moisture content in a soil l ayer

as thick as a few tenths of the wavelength (Chapter 2 and Chapter 4), and thus

provide updating information over a greater depth, being the observation depth.

The simplest way to make a Kalman-filter update of the hydrologic model

from Chapter 5, with near-surface soil moisture data from remote sensing

observations, is to directly apply the inferred soil matric head over the observation

depth d. This eliminates the need for linearisation of the backscattering/brightness

temperature, dielectric and water retention models, thus eliminating linearisation

errors in the observation equation. This is the only way in which model updates

a)0 10 20

0

0.5

1

Day

Eva

pora

tion

(cm

/day

)

b)0 10 20

-400

0

400

Day

Soi

l Hea

t Flu

x (la

ngle

y/da

y)

Figure 6.1: Surface boundary conditions: a) moisture flux boundary condition; and b) heat fluxboundary condition.


may be made using the hard-updating assimilation scheme. Hence, the Kalman-

filtering observation equation (3.3) used in this thesis is

( )v+

î

=

î

−

−

N

N

N

N

d

d

T

T

T

TT

1

2

1

1

2

1

1

2

1

000100000

000000100

000000010

000000001

�

�

�

��

��

��

��

�

ψψ

ψ

ψψ

ψ

ψψ

(6.1),

where T1 is the surface soil temperature from thermal infra-red observations and

ψd is the soil matric suction at the observation depth. Writing the observation

equation in this form makes the soil moisture and temperature profile estimation

algorithm using the Kalman-filter assimilation scheme extremely versatile, as it

allows for updating to be undertaken for either: (i) direct measurements of near-

surface soil moisture content or; (ii ) by inverting any algorithm which relates

near-surface soil moisture content to remote sensing observations. This is a key

difference from the Kalman-filter study presented by Entekhabi et al. (1994),

which linearised the brightness temperature model (2.15) of Njoku and Kong

(1977).

6.3 INITIALISATION PHASE

The one-dimensional soil moisture and heat transfer model from Chapter 5

was initialised with the same poor initial guess used by Entekhabi et al. (1994),

matric head of −300 cm and soil temperature of 15°C uniform throughout the soil

profile. The initial profile conditions are ill ustrated in Figure 6.2.

The same model parameters, boundary conditions and initial profiles as in

Entekhabi et al. (1994), were used to replicate their situation for comparison. It is

then possible to comment on the form of the observation equation used in this


study, and contrast it to the linearisation of the brightness temperature model used

in Entekhabi et al. (1994).

6.4 DYNAMIC PHASE

Upon initialisation with the poor initial guess of the soil moisture and

temperature profiles, the dynamic modelli ng phase was commenced

(section 3.5.2). In this phase, the same boundary conditions as used for generating

the “true” soil moisture and temperature profiles were applied. The one-

dimensional soil moisture and heat transfer model was then updated with near-

surface “observations” (taken from the “true” profile simulations) for various soil

moisture observation depths and updating frequency, using the various

assimilation schemes. This allowed evaluation of the assimilation schemes

eff iciency, and assessment of the effect from observation depth and update

frequency on profile estimation.

6.4.1 CONTINUOUS DIRICHLET BOUNDARY CONDITION

The continuous Dirichlet boundary condition is where the boundary

node(s) are held at a known value for a specified period. When the simulated near-

surface soil states are updated using either the hard-update or Kalman-filter

assimilation schemes, the model estimate of the states in the near-surface soil

layers are replaced with the observations, and the information contained in these

observations is transferred to deeper depths through the physics of the model.

Thus, these updating approaches are in some degree similar to the continuous

a)-600 -300 0

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0

Matric Head (cm)

Dep

th (

cm)

PoorInitialGuess

TrueInitialProfile

b)0 15 30

-100

0

Soil Temperature (oC)

Dep

th (

cm)

PoorInitialGuess

TrueInitialProfile

Figure 6.2: Initial profile conditions: a) soil moisture; and b) soil temperature.


Dirichlet boundary condition, at least for the period during which the updated

surface node(s) remain close to the observation.

To identify how much of the profile estimation was a result of the actual

updating of the model and how much was a result of the continuous Dirichlet

boundary condition, a continuous Dirichlet boundary condition was applied to the

near-surface soil nodes for various observation depths. For the soil moisture

equation, the Dirichlet boundary condition was applied to depths of 0 (surface

node), 1, 4 and 10 cm, while for the temperature equation, the Dirichlet boundary

condition was applied only to the surface node. Observations over a depth of 1 cm

would be representative of what can be achieved from most current remote

sensing systems, whilst 10 cm represents the maximum observation depth that is

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Figure 6.3: Comparison of soil moisture profile estimation using the continuous Dirichletboundary condition for observation depths of 0 (open circle), 1 (open square), 4 (open triangle)and 10 cm (open diamond) with the “true” soil moisture profile (solid circle) and the open loopsoil moisture profile (open circle with dot).


li kely from current technology. Temperature observations are for the surface node

alone, as this is all that can be observed with thermal infra-red sensors.

The soil moisture and temperature profile estimation results are given in

Figure 6.3 and Figure 6.4 respectively. In these figures, the “true” profiles are

compared with the estimated profiles, as well as an “open loop” simulation. The

open loop simulation is where no “observations” were used and the system was

simply propagated from the initial conditions subject to the surface flux

(Neumann) boundary conditions.

The estimated soil temperature profiles differ slightly for the different soil

moisture observation depths, due to the difference in soil moisture profile

estimation. This is a result of the dependence of soil thermal conductivity and heat

capacity on the soil moisture, and the liquid mass flux. The estimated soil

temperature profile plotted in Figure 6.4 is only for the 4 cm soil moisture

observation depth, which is typical of the soil temperature results from other soil

moisture observation depths.

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Figure 6.4: Comparison of soil temperature profile estimation using the continuous Dirichletboundary condition for observations of the surface node (open circle) with the “true” soiltemperature profile (solid circle) and the open loop soil temperature profile (open circle with dot).


The soil moisture content and soil temperature at the surface of the soil

column change with time. In order to maintain the continuous Dirichlet boundary

condition at the “true” value, it was necessary to update the near-surface values

regularly. Therefore the near-surface values used for the continuous Dirichlet

boundary condition were updated once every hour, with the “observed” value at

the beginning of the one hour simulation period. Hence, the simulated soil

temperature in Figure 6.4 always lagged behind the “true” soil temperature by one

hour at the soil surface (see Figure 6.5).

The continuous Dirichlet boundary condition applies the “true” soil

moisture and temperature values near the soil surface. Figure 6.3 and Figure 6.4

show how this boundary condition was slowly transferred from the near-surface to

deeper depths by the model physics. Thus, starting from the soil surface, the

profile estimation algorithm adjusted its estimate of the soil moisture and

temperature profile towards the “true” soil moisture and temperature profile

values, while the open loop showed no improvement. Since the system was drying

and the poor initial guess was somewhat drier than that for the “true” profile, the

open loop moisture profile continued to dry, diverging from the “true” profile.

The open loop soil temperature and moisture profiles responded

differently to the soil moisture profile. As the system was subject to a diurnal soil

heat flux, there was a diurnal soil temperature variation in the upper section of the

soil profile, which continued to maintain its difference from the “true” soil

temperature profile of approximately 5°C, except for very shallow depths. This

difference near the soil surface was most likely due to the open loop soil moisture

0

10

20

30

40

50

60

0 4 8 12 16 20 24

Soi

l Tem

pera

ture

(o C)

Time (h)

Figure 6.5: Illustration of the continuous Dirichlet boundary condition applied for estimation of thesoil temperature profile. The dashed line is the “true” Dirichlet boundary condition while the solidline is the approximation to the Dirichlet boundary condition


profile becoming very much drier than the “true” soil moisture profile at the soil

surface, which allowed greater temperature variations for the same soil heat flux,

as a result of a reduced heat capacity of the soil .

When the Dirichlet boundary condition was applied over deeper depths,

improvements in estimation of the soil moisture profile occurred more quickly.

Soil moisture profile estimates coincided with the “true” soil moisture profile

(ie. the “true” soil moisture profile was retrieved) after approximately 8 days for

the Dirichlet boundary condition at the surface node. In comparison, the “true”

soil moisture profile was retrieved after only approximately 5 days when updating

the top 10 cm. Once the “true” soil moisture profile was retrieved, the model

continued to track the “true” soil moisture profile. This was because there was no

“model error” in these simulations, and the only reason for difference between the

estimated, open loop, and “ true” soil moisture profiles during early simulations,

was the poor initial guess of the soil moisture profile.

Retrieval of the “true” soil temperature profile proceeded more slowly

than for soil moisture, requiring more than 20 days. After 20 days, the estimated

and “ true” soil temperature profiles differed by less than 1°C at deeper depths.

6.4.2 UPDATING ONCE EVERY HOUR

As a first step towards more practically realistic one-dimensional soil

moisture and temperature profile estimation, and to compare with Entekhabi et al.

(1994), both the hard-update and Kalman-filter assimilation schemes were applied

once every hour using surface “observations” with various observation depths.

For the soil moisture equation, the observation depths were taken to be 0 (surface

node), 1, 4 and 10 cm, while for the temperature equation the observation depth

was taken as the surface node. The values that have been used for the surface

“observations” were the simulation values from the “true” profiles for that time

and depth.

This section looks at soil moisture and temperature profile estimation

results from both zero moisture and heat flux at the base of the soil column

(normal simulation) and gravity drainage and heat advection at the base of the soil

column, for comparison with simulation results of Entekhabi et al. (1994). A


sensitivity analysis of soil moisture and temperature profile estimation to factors

likely to influence the difference in Kalman-filter estimates of the soil moisture

profile from those of Entekhabi et al. (1994) is also presented.

6.4.2.1 Normal Simulation

The normal simulation results include assimilation of near-surface

observations using both the hard-updating and Kalman-filtering schemes. Hard-

updating simulation results for soil temperature profile estimation showed littl e

variation with soil moisture updating for different depths. Hence, the soil

temperature profile estimation results are only given for hard-updating of the soil

moisture profile with an observation depth of 4 cm. However, soil temperature

profiles from Kalman-filtering are given for all soil moisture observation depths,

This is because the Kalman-filter produced different soil temperature profile

estimates for different soil moisture observation depths, as a result of its abilit y to

make adjustments to the entire temperature profile, rather than just the surface

node.

Hard-updating results are for the time step immediately prior to the update,

whilst the profiles from the Kalman-filter assimilation scheme are the actual

Kalman-filter update. This is the situation for all simulations presented in this

thesis.

6.4.2.1.1 Hard-Updating

The hard-updating assimilation scheme performs an instantaneous

replacement of the model estimate with the “true” soil moisture and temperature

values over the observation depth once every hour. Thus, the only way in which

extra soil moisture mass or heat energy could be added to or removed from the

soil system was through the observations at the surface node(s). It can be seen

from both Figure 6.6 (soil moisture) and Figure 6.7 (soil temperature) that if this

information was only provided for the surface node, then there was no

improvement in the profile estimates and the results were similar to those of the

open loop. This was most likely because the model was driven by gradients, and

the gradients at the nodes below the surface node over-rode the update, rapidly

replacing the update value with the original value. However, when the time step


size of the very first time step after the update was increased by three orders of

magnitude, then some of the updating information at the surface node was passed

to deeper depths.

The slight variation in the temperature profile from the open loop profile

for the hard-update assimilation scheme was probably a result of the improvement

in soil moisture profile estimates. This was because soil heat capacity and soil

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Figure 6.6: Comparison of soil moisture profile estimation using the hard-update assimilationscheme for observation depths of 0 (open circle), 1 (open square), 4 (open triangle) and 10 cm(open diamond) with the “true” soil moisture profile (solid circle) and the open loop soil moistureprofile (open circle with dot).


thermal conductivity are a function of soil moisture, and the soil temperature

equation is a function of soil moisture content and the soil moisture flux.

Figure 6.6 shows that improvement in soil moisture profile estimation

proceeded more quickly as the observation depth was increased, as was the case

for the continuous Dirichlet boundary condition. However, the effect was much

more pronounced here. This may be because the continuous Dirichlet boundary

condition essentially controlled the rate of moisture flux near the soil surface, with

the depth over which the continuous Dirichlet boundary condition was applied

controlli ng the depth at which this flux was applied. Thus for deeper observation

depths the flux was applied deeper within the soil column, resulting in a slightly

reduced distance for propagation of this boundary condition into the soil profile,

and hence a slightly faster improvement in soil moisture profile estimation. In

hard-updating there was an update of the surface nodes for a given instant in time.

Hence the “true” near-surface moisture flux was only applied for a short period of

time, as this update information was redistributed to deeper depths relatively

quickly, with the near-surface states and fluxes returning quickly to their original

values. More importantly for the hard-update assimilation scheme, is that the

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Figure 6.7: Comparison of soil temperature profile simulation using the hard-update assimilationscheme for observations of the surface node (open circle) with the “true” soil temperature profile(solid circle) and the open loop soil temperature profile (open circle with dot).


depth of update controlled the amount of soil moisture and heat energy that was

added to the system for redistribution to deeper depths. It was this limited

addition/subtraction of soil moisture mass and heat energy to the soil system in

the hard-update assimilation scheme that made the effect of observation depth so

pronounced.

Retrieval of the “true” soil moisture profile using hard-updating

(Figure 6.6) required more than 20 days for observations over 1 cm,

approximately 12 days for 4 cm and approximately 8 days for 10 cm. These

retrieval times are significantly longer than for the continuous Dirichlet boundary

condition of equivalent observation depth, especially for shallower observations,

indicating that the extra soil moisture being added to the system through the

update was playing a more prominent role than the effective Dirichlet boundary

condition.

6.4.2.1.2 Kalman-Filtering

The Kalman-filter performed an instantaneous update of the entire profile

once every hour, based on the relative magnitudes of the covariances of the

observations and the model prediction. It could thus add or subtract soil moisture

mass or heat energy from the system for more than just the near-surface nodes.

However, the values assigned to the initial state covariance matrix, observation

noise and system noise have a significant effect on the improvement in profile

estimation. Hence for comparison with Entekhabi et al. (1994), their observation

and model covariances, and system noise, should be replicated. Entekhabi et al.

(1994) used an observation noise of 2% of the observed state (diagonal) and a

system noise of 5% of the simulated system state (diagonal). However, the value

assigned to the initial state covariances was not stated.

In this study, the initial state covariance matrix was 1000000 for the

diagonal elements and zero for the off diagonal elements, representing a large

uncertainty in the initial profile values and no correlation between nodes. The

observation variances were 2% of the observed system state (matric head or soil

temperature) for the diagonal elements and zero on the off diagonal elements. The

system noise was 5% of the change in system states for each time step, for

diagonal elements, compared with 5% of the actual system state value for


Entekhabi et al. (1994), and zero for the off diagonal elements. The reason for this

was that adding 5% of the state to the diagonal elements of the system covariance

matrix at each time step produced extremely large covariances, and was

dependent on the number of time steps between observations.

Figure 6.8 shows that improvements in the soil moisture profile estimation

proceeded more quickly as the observation depth was increased. However, there

was not a great difference between the time taken to retrieve the “true” soil

moisture profile for observations at the surface node and observations over a

depth of 10 cm.

The updated moisture profile for the very first update at time 1 hour

contained some artefacts, which were not present in later updates. These artefacts

were a result of the initial state covariances and the poor initial guess. However,

as the profile estimation algorithm proceeded, the state covariance matrix was

“warmed up” and the difference between the forecast surface states and

observations became less, so that a more uniform and systematic progression

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Figure 6.8: Comparison of soil moisture profile estimation using the Kalman-filter assimilationscheme for observation depths of 0 (open circle), 1 (open square), 4 (open triangle) and 10 cm(open diamond) with the “true” soil moisture profile (solid circle) and the open loop soil moistureprofile (open circle with dot); initial state variances 1000000, observation variances 2% ofobservations and system noise 5% of change in states.


towards the “true” soil moisture profile was achieved. As the updating progressed,

the Kalman-filter continued to make adjustments to the soil moisture profile at

deeper depths until the “true” soil moisture profile was retrieved, at which stage

the soil moisture profile estimation algorithm continued to track the “true” soil

moisture profile.

Retrieval of the “true” soil moisture profile using the Kalman-fil ter

assimilation scheme required only approximately 12 hours of updating each hour,

independent of the observation depth. This is compared to 8 days for the hard-

update assimilation scheme with observations over a depth of 10 cm, and no

improvement for observations at the surface node. Retrieval of the “true” soil

temperature profile using the Kalman-filter assimilation scheme (Figure 6.9)

required approximately 2 days of updating, compared with no improvement for

the hard-update assimilation scheme. These simulations show very convincingly

that improvements in profile estimation using the Kalman-filter assimilation

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Figure 6.9: Comparison of soil temperature profile estimation using the Kalman-filter assimilationscheme for observations of the surface node (open symbols) with the “true” soil temperatureprofile (solid circle) and the open loop soil temperature profile (open circle with dot). Estimatedsoil temperature profiles correspond with soil moisture profile estimation for observations of 0(open circle), 1 (open square), 4 (open triangle) and 10 cm (open diamond); initial state variances1000000, observation variances 2% of observations and system noise 5% of change in states.


scheme are a result of profile updating over depths greater than the observation

depth.

Comparing the results in Figure 6.8 with those from Entekhabi et al.

(1994) in Figure 6.10, it is obvious that the soil moisture profiles in their

simulations dried much more quickly than in these simulations. Thus, it would

appear that the boundary conditions applied to their model are somewhat different

to those indicated. Either the evaporation rate indicated was too low, or there was

a non-zero moisture flux boundary condition at the soil base. By undertaking

simulations with various evaporation rates and boundary conditions, it became

apparent that a gravity drainage boundary condition at the base of the soil column

gave the closest comparison of soil moisture profiles with those of Entekhabi et

al. (1994).

6.4.2.2 Gravity Drainage and Heat Advection Simulation

As the soil moisture profiles generated in Figure 6.8 were vastly different

to those of Entekhabi et al. (1994), no conclusive comparison could be made with

Figure 6.10: Entekhabi’s comparison of soil moisture profile estimation using the Kalman-filterassimilation scheme (solid circles) with the “true” soil moisture profile (open circles) and openloop soil moisture profile (open triangle) (Entekhabi et al., 1994)


their results. Thus, the hourly update simulations were repeated for gravity

drainage and heat advection at the base of the soil column.

Comparison of “ true” and open loop moisture profiles in Figure 6.6 (zero

flux at base) with Figure 6.11 (gravity drainage) revealed a large difference. The

soil moisture profiles with zero flux at the column base had an initial wetting up

of lower depths in the profile, whilst the profiles from gravity drainage maintained

the initial soil moisture content at depth. The effect of this was to increase the

curvature of the matric head profile under the gravity drainage boundary

condition.

The “true” soil moisture profile dried out much more quickly under the

gravity drainage boundary condition, whilst the open loop profile dried out only

slightly faster. This was due to the moisture control of the moisture flux at the soil

column base, with the gravity drainage flux being greater for higher soil moisture

contents, due to the greater hydraulic conductivity. Hence the “true” and open

loop profiles did not diverge as quickly under the gravity drainage boundary

condition. In addition, the moisture retention curve is exponential. Thus slight

changes in volumetric soil moisture content had a corresponding large change in

matric head at lower soil moisture contents, exacerbating the difference in “ true”

and open loop matric head profiles for the zero flux boundary condition. This

would indicate that the use of a gravity drainage boundary condition should

improve soil moisture profile estimation results, particularly through the moisture

dependence on the gravity drainage, causing the “true” and open loop profiles to

converge from the base. This also means that in practical applications, the

underlying soil will change the characteristics of the soil moisture profile

estimation process.

6.4.2.2.1 Hard-Updating

In both Figure 6.11 (soil moisture) and Figure 6.12 (soil temperature) it

can be again seen that if hard-updates were made for the surface node alone, then

there was no improvement in the profile estimation and the system continued in

the same fashion as the open loop. As the soil moisture observation depth was

increased, improvements in the soil moisture profile estimation were

accomplished. The effect of increasing the observation depth on estimating the


soil moisture profile was again much more pronounced than for the continuous

Dirichlet boundary condition.

Retrieval of the “true” soil moisture profile using hard-updating required

more than 20 days for observations over 1 cm, approximately 16 days for 4 cm

and approximately 10 days for 10 cm. These times were slightly longer than for

the zero flux boundary condition at the base of the soil column, which is counter

intuitive. As time proceeds, the soil moisture profiles converged at the base of the

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Figure 6.11: Comparison of soil moisture profile estimation using the hard-update assimilationscheme for observation depths of 0 (open circle), 1 (open square), 4 (open triangle) and 10 cm(open diamond) with the “true” soil moisture profile (solid circle) and the open loop soil moistureprofile (open circle with dot). Gravity drainage boundary condition at base of soil column.


soil column, as well as at the soil surface, due to the moisture control on the

gravity drainage flux. This is the most notable difference between the simulations

in Figure 6.6 and Figure 6.11

6.4.2.2.2 Kalman-Filtering

The soil moisture profile estimation results in Figure 6.13 for the Kalman-

filter assimilation scheme have again shown that estimation of the “true” soil

moisture profile proceeded slightly faster as the observation depth was increased.

However, there was littl e time difference between the time required to retrieve the

“ true” soil moisture profile for observations at only the surface node and

observations over a depth of 10 cm. The updated profiles in Figure 6.13 contain

the same artefacts as were seen in Figure 6.8, with retrieval of the “true” soil

moisture profile taking approximately 16 hours, independent of the observation

depth. This is compared with 10 days for the hard-update assimilation scheme

with observations over a depth of 10 cm. As for hard-updating, the time required

to retrieve the “true” soil moisture profile using the Kalman-filter assimilation

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Figure 6.12: Comparison of soil temperature profile estimation using the hard-update assimilationscheme for observations of the surface node (open circle) with the “true” soil temperature profile(solid circle) and the open loop soil temperature profile (open circle with dot). Gravity drainageand advection boundary conditions at base of soil column.


scheme for the gravity drainage was slightly longer than for the zero flux

boundary condition.

Retrieval of the “true” soil temperature profile using the Kalman-filter

assimilation scheme required approximately 2 days of updating (Figure 6.14), in

comparison with no improvement in the estimates soil temperature profile for the

hard-update assimilation scheme. The time required to retrieve the “true” soil

temperature profile using the Kalman-filter for the gravity drainage and advection

boundary conditions was approximately the same as for the zero flux boundary

conditions.

The time taken for the soil moisture profile estimation algorithm to

retrieve the “true” soil moisture profile using the Kalman-filter in Figure 6.13 was

much shorter than that required by Entekhabi et al. (1994) in Figure 6.10, being

approximately 16 hours as compared to approximately 4 days. Furthermore, it

should also be noted that the profile estimation of Entekhabi et al. (1994)

converged towards the “true” profile from the bottom. This is counter-intuitive, as

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Figure 6.13: Comparison of soil moisture profile estimation using the Kalman-filter assimilationscheme for observation depths of 0 (open circle), 1 (open square), 4 (open triangle) and 10 cm(open diamond) with the “true” soil moisture profile (solid circle) and the open loop soil moistureprofile (open circle). Gravity drainage boundary condition at base of soil column, initial statevariances 1000000, observation variances 2% of observation and system noise 5% of change instates.


observations are made near the surface of the profile. Hence, it would be expected

that improvements in the profile estimation would converge towards the “true”

profile from the soil surface, as seen in the simulations so far in this thesis.

However, for updating once every 5 days using extremely small i nitial state

variances (Figure 6.37), improvements in soil moisture profile estimates

converged towards the “true” soil moisture profile from the bottom of the soil

column.

6.4.2.3 Sensitivity Analysis of Kalman-Filtering

There are three key factors that play a prominent role in the Kalman-filter

assimilation scheme. Hence, differences in the soil moisture profile estimation as

compared with Entekhabi et al. (1994) may be a result of differences in their

specification. These factors are: (i) the initial state variances; (ii ) the system noise;

and (iii ) the observation noise. As the initial state variances used by Entekhabi et

al. (1994) were not stated, the initial state variances are likely to be different.

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Figure 6.14: Comparison of estimated soil temperature profiles using the Kalman-filterassimilation scheme for observations of the surface node (open symbols) with the “ true” soiltemperature profile (solid circle) and the open loop soil temperature profile (open circle with dot).Estimated profiles correspond with soil moisture profile estimation for observation depths of 0(open circle), 1 (open square), 4 (open triangle) and 10 (open diamond) cm. Gravity drainage andadvection boundary conditions at base of soil column, initial state variances 1000000, observationvariances 2% of observation and system noise 5% of change in states.


Also, application of the system noise by Entekhabi et al. (1994) is unclear,

meaning that the system noise used was not identical. Furthermore, the

observations used by Entekhabi et al. (1994) for soil moisture profile estimation

were the simulated brightness and thermal infra-red temperatures, and the

observations used in this study were the system states (matric head and soil

temperature) for a given observation depth. Thus, the observation noise, being a

proportion of the observed state value, was different for these two situations. To

identify the effect of these three factors on estimation of the soil moisture profile

using the Kalman-filter, a sensitivity analysis of the initial state variances,

observation noise and system noise was performed, for a moisture observation

depth of 1 cm. The effect of model discretisation was also assessed.

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Figure 6.15: Comparison of soil moisture profile estimation using the Kalman-filter assimilationscheme for an observation depth of 1 cm with initial variances of 0 (open circle), 1 (open square),100 (open upward triangle), 10000 (open diamond) and 1000000 (open downward triangle) withthe “true” soil moisture profile (solid circle) and the open loop soil moisture profile (open circlewith dot). Gravity drainage boundary condition at base of soil column, system noise of 5% of thechange in system state during the time step and observation noise of 2% of the observation.


6.4.2.3.1 Sensitivity to Initial State Variances

The initial state variances specify one’s confidence in the initial state

values. Giving these variances a large value signifies littl e confidence in the initial

state values, allowing the Kalman-filter to make strong updates, providing the

observation variances are comparatively low. The value assigned to the initial

state variances will t herefore have a major influence on the abilit y of the Kalman-

filter to make strong updates. To identify how important this was, simulations

were performed for initial state variances of 0, 1, 100, 10000 and 1000000, with a

system noise of 5% of the change in system state during the time step and

observation noise of 2% of the observed state value. These simulations are given

in Figure 6.15 for soil moisture and Figure 6.16 for soil temperature.

Both Figure 6.15 and Figure 6.16 confirm that updating proceeds more

cautiously as the initial state variances are reduced. However, the estimated soil

moisture profile still proceeded towards the “true” soil moisture profile from the

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Figure 6.16: Comparison of soil temperature profile estimation using the Kalman-filterassimilation scheme for observations of the surface node with initial variances of 0 (open circle), 1(open square), 100 (open upward triangle), 10000 (open diamond) and 1000000 (open downwardtriangle) with the “true” soil temperature profile (solid circle) and the open loop soil temperatureprofile (open circle with dot). Gravity drainage and advection boundary conditions at base of soilcolumn, system noise of 5% of the change in system state during the time step and observationnoise of 2% of the observation.


surface down, unlike the results given by Entekhabi et al. (1994). Retrieval of the

“ true” soil moisture profile required approximately 16 hours, 1 day, 2 days and

more than 4 days, while retrieval of the “true” soil temperature profile required

approximately 2 days, 2 days, 2.5 days and 4 days for initial state variances of

1000000, 10000, 100 and 1 respectively. There was no improvement in profile

estimation for an initial state variance of zero. This was expected, as an initial

state variance of zero suggests that the initial states were known exactly for the

entire soil profile. Hence, information contained in the observations was ignored

as it was considered more likely to be in error than the model estimates of the

system states. An initial state variance of 1 still allowed improvement in the

profile estimates, even though it represented a very low uncertainty in the initial

profile states, albeit at a much slower rate. Retrieval of the “true” profiles was still

achieved for the initial state variance of 1, as this small variance signified to the

filter that there was some uncertainty in the model states, and that some

knowledge may be gained from the observations, provided that the uncertainty in

the observations was not too great.

6.4.2.3.2 Sensitivity to System Noise

In addition to the initial state variances, the system noise has a major

influence on the magnitude of the system state covariances, through the

covariance propagation equation. When model updates are performed by the

Kalman-filter, the system state covariances are reduced to account for the extra

knowledge gained from the observations, based on Bayes theory. Thus, if there is

no system noise term, the system state covariances can eventually go to zero. If

the model was perfect, then this would be satisfactory. However, the majority of

models are far from perfect, especially when linearisation is required. Thus, the

system noise term is included to account for the uncertainty in covariance

propagation caused by errors in state propagation. This has the effect of increasing

the system state covariances during the inter-observation periods.

To quantitatively identify the sensitivity of improvements in soil moisture

and temperature profile estimates to the system noise, simulations were performed

for a system noise of 0, 2, 5, 10 and 20% of the change in system state during

each forecast time step, with an observation noise of 2% of the observed state and


an initial state variance of 100. An initial state variance of 100 was used for

several reasons: (i) it allowed for slower convergence of the estimated profile

towards the “true” profile, which was more representative of the profile estimation

given by Entekhabi et al. (1994); and (ii ) the effects from the different values of

system noise were more apparent for a lower initial state variance. These

simulations are given in Figure 6.17 for soil moisture and Figure 6.18 for soil

temperature.

Both Figure 6.17 and Figure 6.18 have shown that changing the system

noise had a minimal effect on the improvement in soil moisture and temperature

profile estimation for this particular situation, with no noticeable differences over

the range of system noise used. However, this may not be the case for a different

form of system noise, as only relatively small changes were made in the system

states for each time step, resulting in relatively low levels of system noise for each

of the cases simulated. Different forms of system noise could be: (i) a percentage

of the maximum change in state over the entire profile for the time step; or (ii ) a

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Figure 6.17: Comparison of soil moisture profile estimation using the Kalman-filter assimilationscheme for an observation depth of 1 cm having a model noise of 0 (open circle), 2 (open square),5 (open upward triangle), 10 (open diamond) and 20 % (open downward triangle) of the change insystem state during the time step with the “true” soil moisture profile (solid circle) and the openloop soil moisture profile (open circle with dot). Gravity drainage boundary condition at base ofsoil column, observation noise of 2% of the observation and initial covariances of 100.


percentage of the actual state value, normalised to account for time step size. In

this way, the system state covariances would not be affected by the number of

time steps taken between observations.

6.4.2.3.3 Sensitivity to Observation Noise

Apart from system state covariances, the rate of improvement in profile

estimation could be influenced by the observation noise. The reason for this is that

Kalman-filter updates are made based on the relative magnitudes of the system

state and observation covariances. Thus, if the system state observation

covariances are large relative to the observation covariances, the Kalman-filter

places its faith in the observations over the model states, making large

adjustments to the model profiles. If on the other hand the observation

covariances are large in comparison to the modelled system state covariances, the

Kalman-filter places its faith in the model states over the observations, making

only minor adjustments to the model profiles. To quantitatively identify the

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Figure 6.18: Comparison of soil temperature profile estimation using the Kalman-filterassimilation scheme for observations of the surface node having a model noise of 0 (open circle),2 (open square), 5 (open upward triangle), 10 (open diamond) and 20% (open downward triangle)of the change in system state during the time step with the “true” soil temperature profile (solidcircle) and the open loop soil temperature profile (open circle with dot). Gravity drainage andadvection boundary conditions at base of soil column, observation noise of 2% of the observationand initial variances of 100.


sensitivity of improvements in profile estimation to the observation covariances,

simulations were performed for an observation noise of 0, 2, 5, 10 and 20% of the

observed state, with an initial state variance of 100 and system noise of 5% of the

change in system state during the time step. These simulations are given in

Figure 6.19 for soil moisture and Figure 6.20 for soil temperature.

Both Figure 6.19 and Figure 6.20 have shown that changing the

observation noise had a strong influence on improvements in the estimation of soil

moisture and temperature profiles, with the rate of improvement in profile

estimation becoming slower as the observation noise was increased. This agrees

with expectations. The rate of improvement in profile soil moisture estimates

occurred much more quickly for “perfect” observations than for even a very low

level of observation noise (2%), with retrieval of the “true” soil moisture profile

requiring approximately 12 hours of updating as compared to 2.5 days. This was

not the case for estimation of the soil temperature profile, with retrieval of the

“ true” soil temperature profile requiring approximately 2.5 days of updating for

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Figure 6.19: Comparison of soil moisture profile estimation using the Kalman-filter assimilationscheme for an observation depth of 1 cm with an observation noise of 0 (open circle), 2 (opensquare), 5 (open upward triangle), 10 (open diamond) and 20% (open downward triangle) of theobservation with the “ true” soil moisture profile (solid circle) and the open loop soil moistureprofile (open circle with dot). Gravity drainage boundary condition at base of soil column, systemnoise of 5% of the change in system state during the time step and initial variances of 100.


both situations, and 3.5 days for higher levels of observation noise. However,

temperature profile updating with “perfect” observations caused some erratic

updating at deeper depths, whilst the “noisy” observations proceeded cautiously in

a monotonic manner towards the “true” profile.

Using a very low initial state variance of 100, system noise of 5% of

change in states during the time step and observation variance of 5% of

observations, resulted in a soil moisture retrieval time that was similar to that of

Entekhabi et al. (1994). However, the way in which the estimated soil moisture

profile approached the “true” soil moisture profile was not consistent with their

results.

6.4.2.3.4 Sensitivity to Model Discretisation

Another factor that could possibly have an influence on the Kalman-filter

update is the number of nodes in the observation depth. Therefore, for soil

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Figure 6.20: Comparison of soil temperature profile estimation using the Kalman-filterassimilation scheme for observations of the surface node with an observation noise of 0 (opencircle), 2 (open square), 5 (open upward triangle), 10 (open diamond) and 20% (open downwardtriangle) of observation with the “true” soil temperature profile (solid circle) and the open loop soiltemperature profile (open circle with dot). Gravity drainage and advection boundary conditions atbase of soil column, system noise of 5% of the change in system state during the time step andinitial variances of 100.


moisture updating over a depth of 1 cm, the number of nodes in the top 1 cm of

the soil profile was increased from 3 to 10. The results from this simulation are

given in Figure 6.21 for soil moisture and Figure 6.22 for soil temperature.

Comparison of Figure 6.21 and Figure 6.22 with Figure 6.13 and

Figure 6.14 indicate that increasing the number of nodes within the observation

depth had a minimal effect on estimation of the “true” soil moisture and

temperature profiles. These simulations indicate a low sensitivity of the rate of

improvement in soil moisture and temperature profile estimation and the shape of

these profiles to the number of nodes in the observation depth.

6.4.3 UPDATING ONCE EVERY DAY

An observation frequency of once every hour is unrealistic for any

practical application of soil moisture profile estimation from remotely sensed

near-surface soil moisture observations. At best, we may expect a repeat coverage

of once every day. However, the results from updating once every hour are

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Figure 6.21: Comparison of soil moisture profile estimation using the Kalman-filter assimilationscheme for an observation depth of 1 cm (open circle) with the “ true” soil moisture profile (solidcircle) and the open loop soil moisture profile (open circle with dot) for an increased number ofnear-surface nodes. Gravity drainage boundary condition at base of soil column, system noise of5% of the change in system state during the time step, observation noise of 2% of the observationand initial state variances of 1000000.


applicable to soil moisture profile estimation from a single soil moisture probe

that is monitoring near-surface soil moisture content at a weather station.

To assess the viabilit y of estimating soil moisture and temperature profiles

from daily observations, both the hard-update and Kalman-filter assimilation

schemes were applied for near-surface observations once every day. This study

was undertaken in the same manner as for updating once every hour, with the

boundary condition at the base of the soil column being zero moisture and heat

flux. This is the boundary condition used for all of the simulations that follow in

this chapter.

6.4.3.1 Hard-Updating

The soil moisture and soil temperature profile estimation results using the

hard update assimilation scheme for observations once every day are given in

Figure 6.23 for soil moisture and Figure 6.24 for soil temperature. The results in

Figure 6.23 indicate once more the benefit of an increased depth knowledge with

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Figure 6.22: Comparison of soil temperature profile estimation using the Kalman-filterassimilation scheme for observations of the surface node (open circle) with the “true” soiltemperature profile (solid circle) and the open loop soil temperature profile (open circle with dot)for an increased number of near-surface nodes. Gravity drainage and advection boundaryconditions at base of soil column, system noise of 5% of the change in system state during the timestep, observation noise of 2% of the observation and initial state variances of 1000000.


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Figure 6.23: Comparison of soil moisture profile estimation using the hard-update assimilationscheme over depths of 1 (open circle), 4 (open square) and 10 cm (open triangle) with the “true”soil moisture profile (solid circle) and the open loop soil moisture profile (open circle with dot).

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Figure 6.24: Comparison of soil temperature profile estimation using the hard-update assimilationscheme for observations of the surface node (open circle) with the “true” soil temperature profile(solid circle) and the open loop soil temperature profile (open circle with dot).


hard-updating. An observation depth of 10 cm facilit ated improvement in the soil

moisture profile estimation much more rapidly than for 4 cm, whilst an

observation depth of 1 cm resulted in essentially no improvement after 20 days.

This was due to the soil moisture mass balance problem discussed in the previous

section.

An observation depth of 1 cm is representative of what can be achieved

from most current remote sensing systems. Hence, it is obvious that a means of

allowing extra soil moisture mass to be added to or subtracted from the system (as

the case may be) must be identified, in order for the hard-update assimilation

scheme to be effective for realistic observation depths and observation intervals.

Figure 6.24 showed once more the failure to make improvements to the soil

temperature profile estimation from an instantaneous hard-update for only the

surface node. Thus for soil temperature profiles to be estimated correctly under

the hard-update assimilation scheme, a means for allowing extra heat energy to be

added to or subtracted from the system was also required.

6.4.3.2 Hard-Updating and Dirichlet Boundary Condition

The most obvious solution to the soil moisture mass and heat energy

balance problem with the hard-update assimilation scheme was to perform the

hard-update and then hold the update values fixed for some period of time.

Providing it is not raining, both soil moisture and temperature values do not

change much over a period of 1 hour. Thus, the hard-update assimilation scheme

was applied with a Dirichlet boundary condition for a period of 1 hour. The

results from this simulation are given in Figure 6.25 for soil moisture and Figure

6.26 for soil temperature. This simulation differs from the continuous Dirichlet

boundary condition simulations in section 6.4.1 in that the Dirichlet boundary

condition was only applied for 1 hour out of every 24 hour period.

Figure 6.25 shows that use of the Dirichlet boundary condition improves

estimation of the soil moisture profile, and that improvements in soil moisture

profile estimation still occurred more quickly for deeper observation depths.

Improvements in soil moisture profile estimation using the 1 cm observation

depth were quicker than improvements in soil moisture profile estimation using

the 4 cm observation depth without the Dirichlet boundary condition.


Improvements in soil moisture profile estimation for the 4 cm observation depth

were approximately equivalent to improvements in soil moisture profile

estimation for the 10 cm observation depth without the Dirichlet boundary

condition. Furthermore, improvements in estimation of the soil moisture profile

for the 4 cm observation depth were approximately twice as fast as for the 1 cm

observation depth, and the 10 cm observation depth was approximately twice as

fast again.

By using the Dirichlet boundary condition for a period of 1 hour after the

update, improvements in estimation of the soil moisture profile were achieved for

all observation depths. However, retrieval of the “true” soil moisture profile did

not occur within 20 days. Likewise, improvements in estimation of the soil

temperature profile were achieved from observation of the soil skin temperature,

but retrieval of the “true” soil temperature profile did not occur within the

20 days. Hard-updating once every day, even with a Dirichlet boundary condition

for 1 hour period, yielded significantly slower retrieval of the “true” soil moisture

profile than an instantaneous hard-update once every hour (8 days as compared to

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Figure 6.25: Comparison of soil moisture profile estimation using the hard-update assimilationscheme with a Dirichlet boundary condition for 1 hour over observation depths of 1 (open circle),4 (open square) and 10 cm (open triangle) with the “true” soil moisture profile (solid circle) andthe open loop soil moisture profile (open circle with dot).


more than 20 days for the 10 cm observation depth). This indicates that more

frequent observations are more useful for estimation of soil moisture and

temperature profiles than knowledge of the near-surface observations for a greater

period of time.

6.4.3.3 Kalman-Filtering

The results from simulations using the Kalman-filter assimilation scheme

are given in Figure 6.27 for soil moisture and Figure 6.28 for soil temperature. In

these figures, the assimilation of near-surface soil moisture and temperature

observations for different soil moisture observation depths have different initial

state variances. The reason for this was that the Kalman-filter assimilation scheme

with daily updating was found to be more sensitive to the initial state variance for

deeper observations.

For observation depths of 4 cm and 10 cm and an initial state variance of

1000000 (Figure 6.29), the Kalman-filter initially made a very poor update

(negative improvement) of the soil moisture profile at the first update (day 1). At

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Figure 6.26: Comparison of soil temperature profile estimation using the hard-update assimilationscheme with a Dirichlet boundary condition for 1 hour at the surface node (open circle) with the“ true” soil temperature profile (solid circle) and the open loop soil temperature profile (open circlewith dot).


the next update (day 2), the Kalman-filter over-corrected for the previous poor

update. This was also observed for estimation of the soil temperature profile with

updating once every hour and zero observation noise (Figure 6.20), but the effect

was not as severe.

To overcome this erratic updating of the soil moisture profile, the initial

state variances were reduced until stable updates were obtained. The initial state

variances required for this were 10000 for the observation depth of 4 cm and 1000

for 10 cm. Even though different initial state variances were used for the deeper

observations, retrieval of the “true” soil moisture profile was achieved after

approximately 3 days for all observation depths. However, improvement in soil

moisture profile estimation for updating at the surface node alone occurred more

quickly than for an observation depth of 1 cm, even though the same initial state

variances were used. Retrieval of the “true” soil temperature profile occurred after

approximately 6 days for all simulations. The different initial state variance

appeared to have littl e effect on the improvements in estimation of the soil

temperature profile, unlike the soil moisture profile. This may be due to the larger

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Figure 6.27: Comparison of soil moisture profile estimation using the Kalman-filter assimilationscheme for observation depths of 0 (open circle), 1 (open square), 4 (open triangle) and 10 cm(open diamond) with the “true” soil moisture profile (solid circle) and the open loop soil moistureprofile (open circle with dot); initial state variances of 1000000, 1000000, 10000 and 1000respectively. Observation variances 2% of observations and system noise 5% of change in states.


coeff icient of variation for the soil temperature profile than for the soil moisture

profile. These results are to be compared with more than 20 days for retrieval of

the “true” soil moisture and temperature profiles using the hard-updating and a

Dirichlet boundary condition.

As for hourly updating, the Kalman-filter assimilation scheme was

superior to the hard-update assimilation scheme, both in terms of time required for

retrieval of the “true” soil moisture and temperature profiles, and the effect of

observation depth on improvements in soil moisture profile estimation. This

highlights the benefit of the non mass-conservative nature of the Kalman-filter

assimilation scheme, by making adjustments to the profile predictions for more

than just the near-surface layer. However, unrealistic updating can occur with the

Kalman-filter when observations become less frequent, the observed and

modelled profiles are far apart, and there is large uncertainty in the model states.

This again emphasises the importance of frequent observations.

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Figure 6.28: Comparison of soil temperature profile estimation using the Kalman-filterassimilation scheme for observations of the surface node (open symbols) with the “ true” soiltemperature profile (solid circle) and the open loop soil temperature profile (open circle with dot).Estimated soil temperature profiles correspond with soil moisture profile estimation forobservations of 0 (open circle), 1 (open square), 4 (open triangle) and 10 cm (open diamond);initial state variances of 1000000, 1000000, 10000 and 1000 respectively. Observation variances2% of observations and system noise 5% of change in states.


6.4.4 UPDATING ONCE EVERY FIVE DAYS

To test how infrequently near-surface observations could be made and still

make satisfactory estimates of the soil moisture and temperature profiles, updating

once every 5 days was examined.

6.4.4.1 Hard-Updating and Dirichlet Boundary Condition

Simulations using the hard-updating assimilation scheme with

observations once every day showed the necessity for identifying a means of

allowing extra soil moisture mass and heat energy to be added to or subtracted

from the system, than that which the hard-updating could achieve on its own. In

the simulations using daily observations, the addition or subtraction of soil

moisture mass and heat energy was achieved by application of the Dirichlet

boundary condition for 1 hour after the update. For 5 day updating, hard-updating

alone would still be of littl e benefit in making improved estimates of the soil

moisture and temperature profiles. Therefore, hard-updating was applied with a

Dirichlet boundary condition for 1 hour after the update.

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Figure 6.29: Comparison of soil moisture profile estimation using the Kalman-filter assimilationscheme with observation depths of 4 (open circle) and 10 cm (open square) with the “ true” soilmoisture profile (solid circle) and the open loop soil moisture profile (open circle with dot); initialstate variance of 1000000. Observation variances 2% of observations and system noise 5% ofchange in states.


6.4.4.1.1 Dirichlet Boundary Condition for One Hour

The hard-updating results with a Dirichlet boundary condition for 1 hour

after updating are given in Figure 6.30 for soil moisture and Figure 6.31 for soil

temperature. The results in Figure 6.30 indicate once again the benefit of an

increased observation depth in the hard-update assimilation scheme. However, the

advantage is much less pronounced than for previous simulations, with retrieval of

the “true” soil moisture profile taking more than 40 days for all observation

depths. Only minimal improvements in the soil moisture profile were achieved for

an observation depth of 1 cm after 40 days. A more satisfactory improvement was

obtained for the soil temperature profile, yet “ true” and estimated soil temperature

profiles still differed by approximately 5°C at depth, after 40 days.

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Figure 6.30: Comparison of soil moisture profile estimation using the hard-update assimilationscheme with a Dirichlet boundary condition for 1 hour over observation depths of 1 (open circle),4 (open square) and 10 cm (open triangle) with the “true” soil moisture profile (solid circle) andthe open loop soil moisture profile (open circle with dot).


6.4.4.1.2 Dirichlet Boundary Condition for One Day

To further alleviate the soil moisture mass and heat energy balance

problem, the Dirichlet boundary condition was applied for a period of 1 day after

the update. The justification for this was that soil moisture does not change by

more than a few percent during a day, unless it is raining. However, the soil

surface temperature has a strong diurnal variation that needs to be accounted for.

Thus, the “true” soil surface temperature values were used for modifying the

Dirichlet boundary condition every hour (see Figure 6.5). The results from this

simulation are given in Figure 6.32 for soil moisture and Figure 6.33 for soil

temperature.

Both Figure 6.32 and Figure 6.33 show once again the advantage of

knowing the “true” near-surface soil moisture and temperature values for a longer

period of time. Using the Dirichlet boundary condition for a period of 1 day,

retrieval of the “true” soil moisture profile was achieved after approximately

40 days for the 10 cm observation depth. Retrieval of the “true” soil temperature

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Figure 6.31: Comparison of soil temperature profile estimation using the hard-update assimilationscheme with a Dirichlet boundary condition for 1 hour at the surface node (open circle), with the“ true” soil temperature profile (solid circle) and the open loop soil temperature profile (open circlewith dot).


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Figure 6.32: Comparison of soil moisture profile estimation using the hard-update assimilationscheme with a Dirichlet boundary condition for 1 day over observation depths of 1 (open circle), 4(open square) and 10 cm (open triangle) with the “true” soil moisture profile (solid circle) and theopen loop soil moisture profile (open circle with dot).

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Figure 6.33: Comparison of soil temperature profile estimation using the hard-update assimilationscheme with a Dirichlet boundary condition for 1 day at the surface node (open circle) with the“ true” soil temperature profile (solid circle) and the open loop soil temperature profile (open circlewith dot).


profile also occurred after approximately 40 days, with “ true” and estimated

profiles differing by approximately 0.5°C at depth. These results are a significant

improvement to those with a Dirichlet boundary condition for only 1 hour after

the hard-update.

6.4.4.1.3 Sensitivity Analysis of the Dirichlet Boundary Condition

The improvement in soil moisture and temperature profile estimates using

the hard-update assimilation scheme was determined by the length of time over

which the continuous Dirichlet boundary condition was applied after the update,

especially when the time between observations was increased. Hence, an

investigation was undertaken to identify if there was a simple relationship

between update interval and length of time for Dirichlet boundary condition, in

order to achieve the same rate of improvement in soil moisture and temperature

profile estimates. To investigate this, the Dirichlet boundary condition was

applied for a fixed proportion of the update interval. Thus, hard updates were

made every 1 day, 2 days and 4 days, with a Dirichlet boundary condition for

1 hour, 2 hours and 4 hours respectively. The results from these simulations are

given in Figure 6.34 for soil moisture and Figure 6.35 for soil temperature.

The simulations in both Figure 6.34 and Figure 6.35 show conclusively

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Figure 6.34: Comparison of soil moisture profile estimation using the hard-update assimilationscheme with a Dirichlet boundary condition over an observation depth of 4 cm (open symbols)with the “true” soil moisture profile (solid circle) and the open loop soil moisture profile (opencircle with dot). Update every day and Dirichlet boundary condition for 1 hour (open circle),update every 2 days and Dirichlet boundary condition for 2 hours (open square), and update every4 days and Dirichlet boundary condition for 4 hours (open triangle).


that the relationship between update interval and the proportion of that interval for

which a Dirichlet boundary condition must be applied in order to achieve the

same rate of improvement in soil moisture and temperature profile estimation

using the hard-update assimilation scheme was not constant. In fact, it was found

that as the interval between observations was increased, knowledge of the “true”

near-surface soil moisture and temperature was required for a greater proportion

of the update interval. This again highlights the greater importance of more

frequent observations then the length of time for which knowledge of the “true”

surface states are available.

6.4.4.2 Kalman-Filtering

Initial simulations with the Kalman-filter assimilation scheme used the

same initial state variance, model noise and observation noise as used in hourly

and daily updating simulations (initial state variance of 1000000, system noise 5%

of change in states and observation noise 2% of observations). This however,

yielded poor updates of the soil moisture profile, as shown in Figure 6.36 for a

1 cm observation depth (open square). Rather than perform an update which lay

somewhere between the “true” and open loop profiles, the updated soil moisture

profile was equal to the “true” soil moisture profile at the soil surface, followed by

an oscill ation between the open loop soil moisture profile and zero matric head,

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Figure 6.35: Comparison of soil temperature profile estimation using the hard-update assimilationscheme with a Dirichlet boundary condition at the surface node (open symbols) with the “true” soilmoisture profile (solid circle) and the open loop soil moisture profile (open circle with dot).Update every day and Dirichlet boundary condition for 1 hour (open circle), update every 2 daysand Dirichlet boundary condition for 2 hours (open square), and update every 4 days and Dirichletboundary condition for 4 hours (open triangle).


before shooting off to a very negative matric head. In subsequent updates there

was a movement of the estimated soil moisture profile towards the “true” soil

moisture profile at deeper depths. However, this movement was slow and the

estimated soil moisture profile was still further from the “true” soil moisture

profile than the open loop profile after 35 days. This is different to that observed

earlier with daily updating, where the Kalman-filter recognised its mistake in the

first update and over-corrected in the second update.

It is well known that the extended Kalman-filter often diverges from the

“ true” solution if the initial estimate is not suff iciently good or the non-linearities

are severe, and that the behaviour of the extended Kalman-filter is often worse

when the model error is large and/or the inputs are small (Ljung, 1979). Hence, in

practical applications of the extended Kalman-filter, “manual” adjustments of the

noise covariances are often used to make the algorithm work, and is referred to as

“ tuning of the filter” (Ljung, 1979). In the daily updating, these stabilit y problems

were overcome by reducing the initial state variance (“ tuning” of the filter).

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Figure 6.36: Comparison of soil moisture profile estimation using the Kalman-filter assimilationscheme for an observation depth of 1 cm with the “true” soil moisture profile (solid circle) and theopen loop soil moisture profile (open circle with dot). Initial state variances of 1000 (open circle),1000000 (open square) and 1e12 (open triangle); system noise 5% of change in states andobservation noise 2% of observations.


Hence, a sensitivity analysis was undertaken to see if the Kalman-filter updates

could be improved by “ tuning” .

6.4.4.2.1 Sensitivity Analysis of the Initial State Variance

The simulation described in the previous section was re-run with an initial

state variance of 1000. The effect of this is shown in Figure 6.36 (open circles),

with the first soil moisture profile update being equal to the “true” soil moisture

profile at the soil surface, followed by an oscill ation towards the open loop profile

before shooting off towards a large positive matric head, which was constrained to

zero. Figure 6.36 also shows the effect of increasing the initial state variance to

1e12 (open triangle), where the first soil moisture profile update was equal to the

“ true” soil moisture profile at the soil surface, before shooting off to a very large

negative matric head. Unlike the simulation with an initial state variance of

1000000, the Kalman-filter realised its mistake at the first update, and over-

corrected at the second update (day 10), with the updated soil moisture profile

being equal to the “true” soil moisture profile at the soil surface, followed by an

oscill ation towards the open loop profile before shooting off towards a large

positive matric head, which was constrained to zero.

By further reducing the initial state covariances, it was possible to obtain

reasonable updates of the soil moisture profile (Figure 6.37). In order to achieve

this, initial state variances were reduced to 10, 5 and 15 for observation depths of

1, 4 and 10 cm respectively. By reducing the initial state variances to such low

values forced the Kalman-filter to perform its updates much more cautiously, as

previously ill ustrated in Figure 6.15. Thus, retrieval of the “true” soil moisture

profile was achieved after approximately 30 days for the 10 cm observation depth.

However, the improvement in soil moisture profile estimates for shallower

observation depths proceeded more slowly, most likely as a result of the smaller

initial state variances. It should also be noted that for these simulations, the

progression in improvement of soil moisture profile estimation towards the “true”

soil moisture profile occurred from the base of the soil column, which is similar to

that observed in the work by Entekhabi et al. (1994).


6.4.4.2.2 Sensitivity Analysis of the Initial Update

Using such extremely small values for the initial state variance indicated

to the Kalman-filter a high degree of confidence in the initial guess, which was

obviously untrue. Furthermore, there was a strong dependence on the initial state

variance for satisfactory updating of the soil moisture profile, resulting in large

differences in the eff iciency of soil moisture profile estimation for slight changes

in the initial state variance.

These two factors initiated a search for a more robust application of the

Kalman-filter assimilation scheme. The first option was to introduce an extra

update shortly after the commencement of the simulation. The reason for this was:

(i) it was felt that if the model prediction was closer to the observation (as with the

soil temperature profile) the updating of the soil moisture profile would be more

robust; (ii ) an early update would move the estimated soil moisture profile closer

to the “true” soil moisture profile (at least in the near-surface soil nodes), resulting

in the model prediction being closer to the observations at the first update; and

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Figure 6.37: Comparison of soil moisture profile estimation using the Kalman-filter assimilationscheme over depths of 1 (open circle), 4 (open square) and 10 cm (open triangle) with the “true”soil moisture profile (solid circle) and the open loop soil moisture profile (open circle with dot).Initial state variances for the estimated soil moisture profiles were 10, 5 and 15 respectively;system noise 5% of change in states and observation noise 2% of observation.


(iii ) in a real li fe application, simulations of soil moisture would commence with

an initial soil moisture profile that was equal to the “true” soil moisture profile for

at least the near-surface nodes, resulting in the model prediction being closer to

the observations at the first update. Thus, preliminary updates were made at time

zero, 1 hour and 1 day.

The results from these simulations are given in Figure 6.38, where it can

be seen that the update at time zero simply replaced the poor initial soil moisture

profile guess with the “true” initial soil moisture profile values over the

observation depth. This was as expected, as there were no cross-correlations in the

initial covariance matrix, which control the updating at deeper depths. The update

at 1 hour made an adjustment to slightly deeper depths in the soil profile, as the

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Day 35.0

Figure 6.38: Comparison of soil moisture profile estimation using the Kalman-filter assimilationscheme for an observation depth of 1 cm with the “true” soil moisture profile (solid circle) and theopen loop soil moisture profile (open circle with dot). First update at time 0 hours (open circle), 1hour (open square) and 1 day (open triangle); initial state variances 1000000, system noise 5% ofchange in states and observation variances 2% of observations.


Kalman-filter had time to build up cross-correlations among the near-surface

nodes. Due to the extra time lapse for the initial update at day 1, the Kalman-filter

was able to make a much larger adjustment to the soil moisture profile. However,

the Kalman-filter still made an over-adjustment to the soil moisture profile

estimate at day 5. The subsequent updates, for preliminary updates at time zero

and 1 hour, show similar characteristics to that of having no preliminary update

(Figure 6.36). However, there was a greater movement towards the “true” profile

than there was for no preliminary update, with the effect being greater for the

preliminary update at day 1. Even after seven updates (day 35) the soil moisture

profile estimate did not agree closely with the “true” soil moisture profile.

6.4.4.2.3 Sensitivity Analysis of the System Noise

Another factor that could affect soil moisture profile updates with the

Kalman-filter was the system noise. When the Kalman-filter makes an update of

the system states, it reduces the magnitude of the covariances of the system state.

Thus for further observations to be beneficial, it is necessary to increase the

system state covariances above the observation noise, during the forecasting

period. The slow improvement of the poor soil moisture profile update with an

initial state variance of 1000000 (Figure 6.36) may be indicative of too littl e noise

in the system. To identify if adding extra noise would improve the soil moisture

profile estimation, three different system noises were investigated: (i) 10% of

change in system states during the time step; (ii ) 5% of the maximum change in

system state over the profile during the time step; and (iii ) 5% of the system state

normalised by the time step size (ie. 5% of the system state per hour).

The results of these simulations are given in Figure 6.39, where it can be

seen that increasing the system noise to 10% of the change in system states during

the time step had a significant effect on the soil moisture profile estimate at the

second update (day 10). The soil moisture profile estimate at day 10 was the same

as for the “true” soil moisture profile at the soil surface, oscill ated towards the

open loop profile, and then back to zero matric head before shooting off to a not

quite so negative matric head at the base of the soil column. The Kalman-filter

then over-corrected at the third update (day 15). The remaining two system noise


scenarios investigated, resulted in the Kalman-filter over-correcting at the first

update.

6.4.4.2.4 Quasi Observations in the Kalman-Filter

Given the failure of all obvious measures (ie. “ tuning” of the filter) for

increasing the robustness of the Kalman-filter assimilation scheme, a new

approach was sought. Near-surface observations of soil moisture content are

indicative of the soil moisture content at depth (see Chapter 3). Thus it was

proposed to apply both the actual observations over the observation depth, and

“quasi” observations to the remainder of the soil profile. This is ill ustrated in

Figure 6.40.

The quasi observations could either be: (i) the observed soil moisture at

the observation depth; or (ii ) an extrapolation of the soil moisture observation at

the observation depth by the steady state assumption. Under this assumption, the

laws of physics state that all points in the soil profile must have the same

hydraulic potential, which is the summation of the hydraulic potential and the

gravitational potential (ie. ψ + z = constant, see also section 3.4.2). It was chosen

to apply the steady state assumption, as this has been shown to be a reasonable

approximation under low flux conditions (Jackson, 1980). In addition, when there

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Figure 6.39: Comparison of soil moisture profile estimation using the Kalman-filter assimilationscheme for an observation depth of 1 cm with the “true” soil moisture profile (solid circle) and theopen loop soil moisture profile (open circle with dot). System noise was 10% of change in states(open circle), 5% of maximum change in states (open square) and 5% of states per hour (opentriangle); initial state variances 1000000 and observation variances 2% of observations.


is a large matric head gradient under exfilt ration conditions, this would have the

effect of making the quasi observations slightly closer to reality.

There is much greater uncertainty associated with a quasi observation then

for the actual observations, even for a layer of soil directly below the observation

depth. With increasing depth from the lowest observation, the uncertainty in the

quasi observation increases dramatically. To account for this, a quantile jump was

applied to the variance of the quasi observation immediately below the

observation depth, relative to the variance of the actual observations. An

increasing quasi observation variance was then applied with depth.

Two scenarios were used to initially test the Kalman-filter assimilation

scheme with quasi observations: (i) quasi observation noise varying from 5% of

the quasi observation to 100% of the quasi observation; and (ii ) quasi observation

noise varying from 5% of the lowest observation to 100% of the lowest

observation. The results from these simulations are given in Figure 6.41, where it

can be seen that both scenarios resulted in a poor estimate of the soil moisture

profile at the first update. At the second update, the soil moisture profile estimate

for the first scenario went to a large negative value at the surface node while the

remainder of the profile went to zero. This was due to the observation variances

being much larger than the system state variances, as a result of insuff icient noise

in the system (5% of change in states), especially at deeper depths where there

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Model

Observations

ModelStandardDeviation

QuasiObservations

QuasiObservation

StandardDeviations

UpdatedProfile

TrueProfile

Figure 6.40: Illustration of the Kalman-filter assimilation scheme using quasi observations.


was not much change in the matric head with time. The updated soil moisture

profile for the second scenario managed to just survive the second update and then

started to become reasonable at the third update. This survival of the second

update was felt to be more good luck than an attribute of the quasi observation

variance. A lack of system noise can also be seen at day 20 and day 25, where the

soil moisture profile estimate for the second scenario did not go to the observed

soil moisture content for the near-surface nodes.

In order to verify that the poor estimates of the soil moisture profile in the

previous simulations were due to a lack of system noise, the simulation was

repeated for a system noise of 5% of the states per hour. In this simulation, only

the second scenario was used, as the first scenario was a misrepresentation of the

desired quasi observation variance. By applying the steady state assumption for

quasi matric head observations, the “observed” matric head is decreasing with

depth. Thus, unless the increase in percentage of state for quasi observations is

large enough, it is possible for a net decrease in quasi observation variance with

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Day 35.0

Figure 6.41: Comparison of soil moisture profile estimates using the Kalman-filter assimilationscheme for an observation depth of 1 cm with the “true” soil moisture profile (solid circle) and theopen loop soil moisture profile (open circle with dot). Quasi observations were applied to theremainder of the profile with observation variances varying from 5% to 100% of the quasiobservation (open circle) and 5% to 100% of the lowest observation (open square); initial statevariances 1000000, observation variances 2% of observations and system noise 5% of change instates.


depth, as was the case with the first scenario. Using the second scenario ensures

that there is a net increase in quasi observation variance with depth. Furthermore,

it makes better sense to determine the variance of the quasi observations a

function of the soil moisture value used to estimate the quasi observation.

The results of this second simulation are given in Figure 6.42, where it can

be seen that the increased system noise resulted in the Kalman-fil ter making a

good estimate of the “true” soil moisture profile after the first update. The soil

moisture profile estimate then continued to follow the “true” soil moisture profile

through to approximately day 25, where it started to move towards the open loop

profile. This can be seen more distinctly at day 30 and day 35, and was a result of

the poor quasi observations for these updates. At these updates, the matric head

was very largely negative at the soil surface and had a steep matric head gradient

with depth, being far from the steady state condition. Thus the quasi observations

at deeper depths were far from the “true” soil moisture profile. As a result of the

quasi observation variances being too small compared to the variances of the

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Day 40.0

Figure 6.42: Comparison of soil moisture profile estimation using the Kalman-filter assimilationscheme for an observation depth of 1 cm with the “true” soil moisture profile (solid circle) and theopen loop soil moisture profile (open circle with dot). Quasi observations were applied to theremainder of the soil profile with observation variances varying from 5% to 100% of the lowestobservation; system noise 5% of change in states (open circle) and 5% of states per hour (opensquare); initial state variances 1000000 and observation variances 2% of observations.


forecast system states, the Kalman-filter placed increasingly more faith in the

quasi observations than in the forecast system states, thus moving its best estimate

towards the quasi observations.

To overcome the problem with poor quasi observations at later updates,

simulations were run with increased variances on the quasi observations. The

variances that were used varied linearly from: (i) 10% of the lowest observation to

200% of the lowest observation; and (ii ) 20% of the lowest observation to 400%

of the lowest observation. These results are compared in Figure 6.43 with the

simulation using a quasi observation variance that varied linearly from 5% of the

lowest observation to 100% of the lowest observation.

These simulations show that with the increased quasi observation variance,

an extra update was required in order to retrieve the “true” soil moisture profile.

However, the increased quasi observation variance alleviated the effect of poor

quasi observations on estimation of the soil moisture profile after the “true” soil

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Day 35.0

Figure 6.43: Comparison of soil moisture profile estimation using the Kalman-filter assimilationscheme for an observation depth of 1 cm with the “true” soil moisture profile (solid circle) and theopen loop soil moisture profile (open circle with dot). Quasi observations are applied to theremainder of the soil profile with observation variances varying from 5% to 100% (open circle),10% to 200% (open square) and 20% to 400% (open triangle) of the lowest observation; initialstate variances 1000000, observation variances 2% of observations and system noise 5% of statesper hour.


moisture profile was correctly retrieved, particularly for the larger quasi

observation variances.

Given a soil moisture profile estimation algorithm that reliably estimated

the soil moisture profile for the 1 cm observation depth, we were in the situation

where we could test it for other observation depths. The results from these

simulations are given in Figure 6.44 for soil moisture and Figure 6.45 for soil

temperature.

These results show that the “true” soil moisture profile was retrieved after

only 10 days (two updates) whilst retrieval of the “true” soil temperature profile

occurred after only 15 days. This is to be compared with 40 days for retrieval of

soil moisture and temperature profiles using the hard-update assimilation scheme

with an observation depth of 10 cm and Dirichlet boundary condition for 1 day,

again showing the advantage of the Kalman-filter. However, the Kalman-filter

was not without its problems. Once retrieval of the “true” soil moisture profile

was achieved, the profile estimation algorithm continued to track the “true” soil

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Day 35.0

Figure 6.44: Comparison of soil moisture profile estimation using the Kalman-filter assimilationscheme for observation depths of 0 (open circle), 1 (open square), 4 (open triangle) and 10 cm(open diamond) with the “true” soil moisture profile (solid circle) and the open loop soil moistureprofile (open circle with dot); quasi observations were applied over the remainder of the soilprofile with variances varying from 20% to 400% of the lowest observation. Initial state variancesof 1000000, observation variances 2% of observations and system noise 5% of states per hour.


moisture profiles until day 30. At this time, estimates of the soil moisture profile

using observations of the surface node began to depart from the “true” soil

moisture profile. This was again caused by the departure of the “true” soil

moisture profile from steady state and the extremely large negative matric head at

the soil surface. Under field conditions the situation would most likely be

somewhat different, as evaporation would not be occurring at a constant rate,

allowing for capill ary rise during periods of low evaporation. Thus, this departure

from the “true” profile at later updates did not warrant any major concern, as this

would be an extreme situation.

The main focus in this section of the synthetic study was to make

satisfactory updates with the Kalman-filter during the initial stages of soil

moisture and temperature profile estimation. The reason this may have been

possible with the quasi observations in these simulations, is that the “true” profile

was approximately steady state during the early updates. Only a field application

will t ruly reveal i f this is the situation. However, providing the model is initialised

5 10 15 20 25 30-100

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Figure 6.45: Comparison of soil temperature profile estimates using the Kalman-filter assimilationscheme for observations of the surface node (open symbols) with the “true” soil temperatureprofile (solid circle) and the open loop soil temperature profile (open circle with dot). Soiltemperature profiles correspond with soil moisture profile estimation for observation depths of 0(open circle), 1 (open square), 4 (open triangle) and 10 cm (open diamond); initial state variancesof 1000000, observation variances 2% of observations and system noise 5% of the states per hour.


at an appropriate time (ie. period of saturation) this should not be an issue for

application of the soil moisture profile estimation algorithm using the Kalman-

filter assimilation scheme.

6.4.4.2.5 Log Transformation in the Kalman-Filter

Whilst it was necessary to apply quasi observations to the unobserved

portion of the soil moisture profile in order to provide stabilit y to Kalman-filter

updates with observations every 5 days, soil temperature profile updates did not

exhibit any of these problems when using only the surface node observations. It

would appear from this that providing the observations are not too far from the

forecast system states, the Kalman-filter can provide a stable update. Thus, if we

can possibly reduce the difference between the observed and forecast soil

moisture values, we may be able to estimate the moisture profile using only the

soil moisture observations over the observation depth. One way in which this

reduction in difference between observed and simulated near-surface soil moisture

can be achieved is through a log transformation of the matric head. This is

ill ustrated in Figure 6.46.

In order to apply this transformation, both the observations and forecast

system states, and their covariances, must be transformed into log space. These

transformations may be achieved through the relationships (Bras and Rodriguez-

Iturbe, 1985)

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Prediction

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TrueProfile

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Dep

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ModelProfile

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TrueProfile

Figure 6.46: Illustration of reduction in difference between observed and measured near surfacesoil moisture using a log transformation of the matric head.


( )2

log

2'

'i

ii

X

XeX

σµµ −= (6.2a)

+= 1log

2

2'

'

i

i

iX

X

eX µ

σσ (6.2b)

{ }( ) { }( )''

''

''

1exp1exp1log

ji

jiji

ji

XX

XXXXe

XX σσ

σσρρ

−−+= (6.2c)

{ }( )'

'

'

1exp 2

i

iji

ji

X

XXX

XX σ

σρρ

−= (6.2d),

where Xi and Xj are the ith and jth states in the original system with mean µ,

standard deviation σ and correlation coeff icient ρ. Xi' and Xj' are the ith and jth

states in the log transformed system. As the Kalman-filter tracks the conditional

mean of the system, the means of the original system are the forecast system

states.

After the system update, the updated system states and their covariances

must be transformed back into the original system. These transformations may be

achieved through the relationships (Bras and Rodriguez-Iturbe, 1985)

î

+= '

'

2exp

2

i

i

i X

X

X µσ

µ (6.3a)

[ ]{ } { }'''' 2exp2exp 222

iiiii XXXXX µσµσσ +−+= (6.3b)


{ }{ } 1exp

1exp

''

''''

−

−=

ji

jiji

ji

XX

XXXX

XX σσ

ρσσρ (6.3c)

{ }( )1exp 2'

''

−=

i

iji

ji

X

XXX

XXσ

σρρ (6.3d).

Using the log transformation, stable updates of the soil moisture profile

were achieved using the 1 cm observation depth without quasi observations, for

an initial state variance of 1000000, system state noise of 15% of the states per

hour, and an observation noise of 2% of the observations. The results from this

simulation are given in Figure 6.47, where it can be seen that the soil moisture

profile estimate coincided with the “true” soil moisture profile after approximately

10 days. Once the “true” soil moisture profile was retrieved, the soil moisture

profile estimation algorithm continued to track the “true” soil moisture profile.

However, the first update (day 5) was very sensitive to the initial state variance

and system state noise, with other values resulting in the same unstable updates

shown in previous sections. In addition, the combination of initial state variance

and system noise required for performing a stable update for other observation

depths could not be identified.

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-60

-40

-20

0

Matric Head (cm)

Dep

th (

cm)

Day 5.0

-1000 -800 -600 -400 -200 0-100

-80

-60

-40

-20

0

Matric Head (cm)

Dep

th (

cm)

Day 10.0

-1000 -800 -600 -400 -200 0-100

-80

-60

-40

-20

0

Matric Head (cm)

Dep

th (

cm)

Day 15.0

-1000 -800 -600 -400 -200 0-100

-80

-60

-40

-20

0

Matric Head (cm)

Dep

th (

cm)

Day 20.0

-1000 -800 -600 -400 -200 0-100

-80

-60

-40

-20

0

Matric Head (cm)

Dep

th (

cm)

Day 25.0

Figure 6.47: Comparison of soil moisture profile estimation using the Kalman-filter assimilationscheme for an observation depth of 1 cm and log transformation (open circle) with the “ true” soilmoisture profile (solid circle) and the open loop soil moisture profile (open circle with dot); initialstate variances 1000000, observation variances 2% of observations and system noise 15% of statesper hour.


6.4.4.2.6 Volumetric Moisture Transformation in the Kalman-Filter

Whilst the log transformation reduced the difference between the

observations and model predictions of near-surface soil moisture, the transformed

soil moisture profile maintained the large gradient of matric head with depth near

the soil surface. However, the corresponding volumetric soil moisture profile does

not exhibit this same property (Figure 6.48), as volumetric soil moisture content is

constrained by the residual soil moisture content and the soil porosity. Thus, a

volumetric soil moisture transformation reduces the difference between

observations and model predictions, as well as the non-linearities in the shape of

the soil moisture profile, particularly in the vicinity of the near-surface

observations. Thus, the non-linearities in the ψ-based model used in

PROXSIM1D (Chapter 5), which are believed to be the cause of the extended

Kalman-filter updates diverging from the “true” soil moisture profile, are reduced

by using a θ-based model.

The transformation of the model soil moisture state from matric head to

volumetric soil moisture content can be achieved by the water retention

relationships in (5.44) to (5.46). However, the covariances of the model states

must also be transformed into volumetric soil moisture space. This may be

achieved by

TBB XY Σ=Σ (6.4),

-3000 -2500 -2000 -1500 -1000 -500 0-100

-80

-60

-40

-20

0

Matric Head (cm)

Dep

th (

cm)

ModelPrediction

Observations

TrueProfile

0 0.1 0.2 0.3 0.4 0.5 0.6-100

-80

-60

-40

-20

0

Soil Moisture (v/v)

Dep

th (

cm) Model

Prediction

Observations

TrueProfile

Figure 6.48: Illustration of the reduction in profile non-linearity by using a volumetric soilmoisture transformation of the soil matric head.


where ΣY is the covariance matrix of Y (the transformed system) and ΣX is the

covariance matrix of X (the original system), being the covariance matrix

requiring transformation. For the general case of the system states ψ and T

augmented with the system parameters α (see Appendix A), then

=

N

N

l

l

l

T

TN

α

α

θ

θθ

�

�

�

1

1

2

1

Y

=

N

N

N

T

T

α

α

ψ

ψψ

�

�

�

1

1

2

1

X (6.5).

The matrix B is a transformation matrix given by

=

m

mmmm

m

m

llll

m

llll

T

T

T

TTT

T

T

∂α∂α

∂∂α

∂ψ∂α

∂ψ∂α

∂α∂

∂∂

∂ψ∂

∂ψ∂

∂α∂θ

∂∂θ

∂ψ∂θ

∂ψ∂θ

∂α∂θ

∂∂θ

∂ψ∂θ

∂ψ∂θ

��

�

��

��

��

121

1

1

1

2

1

1

1

121

121

2222

1111

B (6.6).

The relationship between Y and X is given by

( )αψθ ,jl fj

= (6.7a)

jj TT = (6.7b)


ii αα = (6.7c).

As the transformation of the system states at node j is independent of the

system states at other nodes and the transformation of parameter i is independent

of the other parameters, a diagonal matrix is obtained with zero on the off

diagonal terms. The exception to this is the dependence of system states to the

model parameters in the top right hand quadrant. Thus, the transformation matrix

may be given by

=

10

00

01

10

00

01

0

0

0

1

1

11

1

�

�

��

m

ll

m

ll

NN

NC

C

∂α∂θ

∂α∂θ

∂α∂θ

∂α∂θ

ψ

ψ

B (6.8),

where Cψ is the capill ary capacity factor ∂θ⁄∂ψ given by (5.47) to (5.49) and ∂θ⁄∂α

are given by (A.39) to (A.49) of Appendix A. The re-transformation of the

updated covariance matrix may be achieved by

-1T1 BB YX Σ=Σ − (6.9).

The problem associated with transforming the system states from matric

head to volumetric soil moisture is the assumption of normality for the errors.

That is, when the soil moisture content approaches the residual soil moisture

content or the soil porosity, the Kalman-filter believes the standard deviation

should be small , as soil moisture content cannot be less than the residual soil

moisture content or greater than the soil porosity. What the Kalman-filter does not

recognise is that the forecast soil moisture content could be much wetter in the dry

case, or much drier in the wet case. The problem that this creates is that the


Kalman-filter interprets these small standard deviations as a high degree of

certainty in the model prediction and ignores the observation. To overcome this, a

limit was placed on the minimum value for ∂θ⁄∂ψ to ensure that reasonably large

standard deviations were maintained for the transformed system states near the

soil surface, whilst ensuring that the standard deviation was not greater than the

soil porosity. A value of 1e-6 was used for this purpose.

Using this volumetric soil moisture transformation, stable soil moisture

profile updates were obtained for all observation depths apart from the surface

node. Stable updates could not be achieved for surface node observations, as the

transformation process resulted in a low correlation with forecast soil moisture

content at deeper depths. The results for simulations using the volumetric soil

moisture transformation are given in Figure 6.49.

These results show that the “true” soil moisture profile was retrieved after

10 days for the 10 cm observation depth and 15 days for 1 cm and 4 cm

observation depths. A larger initial state variance was used for the 1 cm

0 10 20 30 40 50 60-100

-80

-60

-40

-20

0

Volumetric Soil Moisture (%)

Dep

th (

cm)

Day 0.0

0 10 20 30 40 50 60-100

-80

-60

-40

-20

0


Dep

th (

cm)

Day 5.0

0 10 20 30 40 50 60-100

-80

-60

-40

-20

0


Dep

th (

cm)

Day 10.0

0 10 20 30 40 50 60-100

-80

-60

-40

-20

0


Dep

th (

cm)

Day 15.0

0 10 20 30 40 50 60-100

-80

-60

-40

-20

0


Dep

th (

cm)

Day 20.0

0 10 20 30 40 50 60-100

-80

-60

-40

-20

0


Dep

th (

cm)

Day 25.0

0 10 20 30 40 50 60-100

-80

-60

-40

-20

0


Dep

th (

cm)

Day 30.0

0 10 20 30 40 50 60-100

-80

-60

-40

-20

0


Dep

th (

cm)

Day 35.0

Figure 6.49: Comparison of soil moisture profile estimation using the Kalman-filter assimilationscheme for observation depths of 1 (open circle), 4 (open square), and 10 cm (open triangle) withthe “true” soil moisture profile (solid circle) and the open loop soil moisture profile (open circlewith dot); moisture transformation of states and state covariances. Initial state variances of1000000, 10000 and 10000 respectively, observation variances 2% of observations and systemnoise 5% of states per hour.


observation depth (1000000) than for the 4 and 10 cm observation depths (10000)

to ensure that large standard deviations were obtained for near-surface nodes after

the transformation. These profile retrieval results can be compared with those

from the quasi observation simulations, which required 10 days for all observation

depths. Only one extra update was required using the volumetric soil moisture

transformation, and no assumption was required regarding the soil moisture

profile. Furthermore, had soil moisture been the dependent state in the soil

moisture model, retrieval of the “true” soil moisture profile may have been

achieved more rapidly, as the transformation of covariances and its associated

problems and assumptions would have been eliminated.

Hence, it would appear obvious that whilst the ψ-based form of the

moisture equation is more correct in terms of modelli ng profile soil moisture for

multi -layered soils, the θ-based form is required for stable updating with the

Kalman-filter when observations and model predictions have a large departure.

The reason for this is that the θ-based form of the moisture equation is more linear

than the ψ-based form.

6.5 CHAPTER SUMMARY

It has been shown that the Kalman-filter assimilation scheme is superior to

the continuous Dirichlet boundary condition and the hard-updating assimilation

schemes, and that there is no improvement made to the estimate of the soil

moisture or temperature profile through making hard-updates of the surface nodes

alone. A summary of the simulation times required for retrieval of the “true” soil

moisture and temperature profiles using the different assimilation schemes is

given in Table 6.2, for the different observation intervals and soil moisture

observation depths.

The superiority of the Kalman-filter comes through its abilit y to make

adjustments to the entire profile, whilst hard-updates can only directly alter the

profile within the observation depth. However, the Kalman-filter can only do this

if there is a high correlation between the states of adjacent nodes. Thus, the model

used for forecasting of the system states must have a dependence on the system

states of the adjacent nodes.


Being unable to directly alter more than the observed soil moisture and

temperature values using the hard-updating assimilation scheme created a soil

moisture mass and heat energy balance problem, as the soil moisture mass and

heat energy added during an instantaneous hard-update is restricted by the depth

Table 6.2: Summary of soil moisture and temperature profile retrieval times from the syntheticstudy using hard-updating and Kalman-filtering, with various observation depths and updateintervals.

Hard-Updating Kalman-Filtering

Soil Moisture Soil Temperature Soil Moisture Soil Temperature

Observation

Depth (cm)0 1 4 10 Surface 0 1 4 10 Surface

Update

Interval Profile Retrieval Time (days) Profile Retrieval Time (days)

Continuous 8 8 7 5 >20

1 Hour >20 12 8 − − 0.5 0.5 0.5 0.5 2

1 Hour1 − >20 16 10 − 0.7 0.7 0.7 0.7 2

1 Day2 >20 >20 >20 >20

1 Day 3 3 3 3 6

5 Days3 >40 >40 40 40

5 Days4 10 10 10 10 15

5 Days5 − 10 − −

5 Days6 − 15 15 10

1. Gravity drainage and advection boundary condition at base of soil column

2. Dirichlet boundary condition at soil surface for 1 hour after update

3. Dirichlet boundary condition at soil surface for 1 day after update

4. Quasi observations applied to remainder of profile

5. Log transformation

6. Moisture transformation


of the observation. This surface information is transferred to deeper depths

through the internal physics of the model (ie. infilt ration/exfilt ration). Thus, an

increased observation depth was an obvious advantage for the hard-update

assimilation scheme. As the observations became less frequent, the hard-update

assimilation scheme required a Dirichlet boundary condition, which had to be

applied for an increasingly longer proportion of the update interval. This indicated

that more frequent observations are more useful for profile estimation than

knowledge of the surface states for a greater period of time.

The soil moisture mass and heat energy added during a continuous

Dirichlet boundary condition is constrained by the physical rate at which soil

moisture and heat can be transferred through the soil profile, and the length of

time for which the Dirichlet boundary condition is maintained. Thus, observation

depth had a reduced influence on the soil moisture and temperature profile

retrieval time when the Dirichlet boundary condition was applied.

It has also been shown that observation depth did not have a significant

effect on the “true” soil moisture and temperature profile retrieval time for the

Kalman-filter assimilation scheme. However, it was observed that unrealistic

updating of the profile occurred with the Kalman-filter when observations become

less frequent, the observed and modelled profiles were far apart, and there was a

large uncertainty in the modelled profiles. This again highlighted the importance

of frequent observations, and suggested that for the Kalman-filter assimilation

scheme, that repeat coverage frequency is more important than observation depth.

Whilst stable updating of the soil moisture profile using the Kalman-filter

assimilation scheme was achieved by applying quasi observations for the

unobserved portion of the moisture profile, this was undesirable and the

usefulness of this approach was not widely verified. However, it has been shown

that the Kalman-filter assimilation scheme was less susceptible to unstable

updates if volumetric soil moisture was modelled as the dependent state, as this

reduced the non-linearities in the soil moisture model. This is a key outcome of

the synthetic study, as it has given invaluable insight relating to the model

structure requirements for the spatial problem.


This synthetic study has shown that to estimate soil moisture profiles using

the water balance approach, an assimilation scheme that has the non-mass

conservative characteristics of the Kalman-filter is essential for eff icient updating

of the soil moisture and temperature profiles. Moreover, when using an

assimilation scheme having this characteristic, the retrieval of “ true” soil moisture

and temperature profiles was insensitive to observation frequency and observation

depth, providing a linear form of the forecasting model was used

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CHAPTER SIX - User Web Pagesusers.monash.edu.au/~jpwalker/thesis_pdf/chapter6.pdf · 2000-08-25 · 6.3 INITIALISATION PHASE The one-dimensional soil moisture and heat transfer model